In this paper, we develop two families of sequential monitoring procedure to (timely) detect changes in a GARCH(1,1) model. Whilst our methodologies can be applied for the general analysis of changepoints in GARCH(1,1) sequences, they are in particular designed to detect changes from stationarity to explosivity or vice versa, thus allowing to check for volatility bubbles. Our statistics can be applied irrespective of whether the historical sample is stationary or not, and indeed without prior knowledge of the regime of the observations before and after the break. In particular, we construct our detectors as the CUSUM process of the quasi-Fisher scores of the log likelihood function. In order to ensure timely detection, we then construct our boundary function (exceeding which would indicate a break) by including a weighting sequence which is designed to shorten the detection delay in the presence of a changepoint. We consider two types of weights: a lighter set of weights, which ensures timely detection in the presence of changes occurring early, but not too early after the end of the historical sample; and a heavier set of weights, called Renyi weights which is designed to ensure timely detection in the presence of changepoints occurring very early in the monitoring horizon. In both cases, we derive the limiting distribution of the detection delays, indicating the expected delay for each set of weights. Our theoretical results are validated via a comprehensive set of simulations, and an empirical application to daily returns of individual stocks.

翻译：本文开发了两类序贯监测程序，用于（及时）检测GARCH(1,1)模型中的变化点。尽管我们的方法可推广应用于GARCH(1,1)序列中变化点的通用分析，但其核心设计目标为检测从平稳性向爆炸性（或反之）的转变，从而能够检验波动率泡沫。我们的统计量可适用于历史样本是否平稳的情形，且无需事先知晓断点前后观测区间的状态。特别地，我们基于似然函数的拟费希尔得分构建CUSUM过程作为检测器。为确保及时检测，我们通过引入赋权序列构建边界函数（超出该函数即表明发生断点），该序列旨在缩短变化点存在时的检测延迟。我们考虑两类权重：轻量级权重集，可确保在历史样本结束后的早期（但非过早）发生变化时实现及时检测；以及称为Rényi权重的重量级权重集，专为监测窗口初期出现变化点时实现及时检测而设计。针对两类权重，我们推导了检测延迟的极限分布，指明各类权重对应的期望延迟时长。通过系统性模拟验证理论结果，并将实证方法应用于个股日度收益率数据。